EMPIRICAL DESCRIPTION OF HIGH WAVE NUMBER TURBULENCE SPECTRA Richard Baker Belser POSTGRADUATE SCHOOL Monterey, California Bw^pw* L_l \ Empirical Description of High Wave Number Turbulence Spectra by Richard Baker Belser III Thesis Advisor: N. E. Boston September 1972 I 1 i, IL 0 ;8 Approved faon public ^.cXeaie; diMtxihution unlimited. Empirical Description of High Wave Number Turbulence Spectra by Richard Baker Belser III Lieutenant Commander, United States Navy B.S., United States Naval Academy, 1964 Submitted in Partial Fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OCEANOGRAPHY from the NAVAL POSTGRADUATE SCHOOL September 1972 ABSTRACT The shapes of spectra obtained from measurements of ocean turbulence are examined. Kovasznay's equation for high wave number turbulence is used to describe the inertial through dissipation regions. Power spectra of velocity fluctuation recorded in the open sea are used to evaluate constants occur- ring in the Kovasznay relation. Semi-empirical forms of high wave number energy spectra are then displayed. These computed spectra should define a universal form to which mathematical models should adhere. TABLE OF CONTENTS I. INTRODUCTION 7 A. BACKGROUND ; 7 B. APPROACH 9 C. KOVASZNAY'S EQUATION 10 1. Theoretical Background-' 10 2. Kovasznay's Proposal 11 3. Application 14 II. DESCRIPTION OF DATA 15 A. DATA COLLECTION 15 1. Area- 15 2. Sensors 16 3. Errors 18 4. Spectra 18 5 . Recording 19 B. DATA USED 20 III. ANALYSIS AND APPROACHES 22 A. APPROACHES 22 1. Linear Regression 22 2. Inertial Subrange 22 3. Exponential Curve 22 B. ANALYSIS 23 1. Linear Regression' 23 2. Substitution ' 24 3. Gompertz Curve 24 IV. CONCLUSIONS 25 TABLE I - VALUES OF A-, A£ , A3 27 APPENDIX A - k and 0(k) VALUES FOR DATA SET 4, . 28 APPENDIX B - k and 0 (k) VALUES FOR DATA SET 5— 29 APPENDIX C - k and 0Ck) VALUES FOR DATA SET 6-. -- 30 APPENDIX D - k and 0(k) VALUES FOR DATA SET 7 31 APPENDIX E - k and 0 (k) VALUES FOR DATA SET 8' 34 APPENDIX F - k and 0(k) VALUES FOR DATA SET 9 36 BIBLIOGRAPHY 4 7 INITIAL DISTRIBUTION LIST 48 FORM DD 1473 49 LIST OF FIGURES 1. Velocity Spectrum Data Set 4 ■ 38 2. Velocity Spectrum Data Set 5 — 39 3. Velocity Spectrum Data Set 6 40 4. Velocity Spectrum Data Set 7 41 5. Velocity Spectrum Data Set 8 42 6. Velocity Spectrum Data Set 9 43 7. Comparison of Non-Normalized Curve (Data Set 4) and Curve Regeneration using A,, A_ , and A„ 44 8. Kovasznay a versus wave number k 45 9. Curve Regeneration using Actual a Values and Kovasznay's Equation 46 ACKNOWLEDGMENT The author would like to acknowledge the generous assist- ance given by Dr. Noel E. Boston and Dr. Kenneth L. Davidson, thesis advisors, and by the Academic Council for granting a thesis extension for this research. The data reported in the appendices were provided by Dr. P. W. Nasmyth, West Coast Division, Marine Sciences Branch, Canada. I. INTRODUCTION A. BACKGROUND Turbulence occurring at small scales of the order of me- ters to centimeters are features that are of interest to engi- neers, oceanographer s and meteorologists. Much of this interest lies in the transport properties of momentum, energy, and heat in turbulence even though the bulk of these fluxes take place over a wide range of scales. At larger scales such as encountered in geophysics the turbulence is governed by mean flow features and hence is influenced by boundary conditions and meteorological and oceanographic variables. At small scales, the turbulence becomes more ordered (in a stochastic sense) as the influence of the mean flow becomes less. Certain proper ties /features are common from one flow to another. At reasonably small scales turbulence is appar- ently governed by "universal" laws. A "universal" law implies that turbulence found in the ocean, atmosphere, or laboratory in gases, liquids, or plasmas will have the same behavior. In 1941, A. N. Kolmogorov set up a mathematical framework for small scale turbulence. A statistical mathematician, Kolmogorov, nonetheless, based his theory on physical reason- ing. As originally formulated, his theory applied to scales of motion which were isotropic. The theory is known as the Kolmogorov theory of isotropic turbulence. This theory pre- dicted results which were subsequently experimentally veri- fied. Specifically Kolmogorov predicted that there was a range of scales where energy would fall off with wave number to the minus five-thirds power. This range where energy is transferred inertially is called the inertial subrange and describes scales of turbulence which are separate from the mean flow but not small enough to be affected by the viscos- ity. In this range 000 = K' e2/3 k"5/3(k> (1) where 0(k) is the one-dimensional energy spectrum ( ) (cm), K ' is a "universal" constant, £ is the rate of dissipa- 1 txon of turbulent kinetic energy ( ) ° sec wave number ( ) . cm sec , and k is the Recent experimental work has shown. that this equation holds over a much wider- range of scales than Kolmogorov anti- cipated. The law holds at large scales where the turbulence cannot possibly be isotropic. This is important to resear- chers in geophysical fluids because they are provided with a law which can be applied to a wide range of geophysical prob- lems . Turbulence in the atmosphere and in the ocean almost always exhibits an inertial subrange. If K' is known in the inertial subrange then e may be determined by measuring 0(k). The evaluation of e is critical in oceanic turbulence and air-sea interaction since if e is known, other parameters such as the drag coefficient, C, and sensible heat flux may be es timated . The "universal" law therefore easily allows the computa- tion of e, the paramount parameter of the theory, provided 8 that K' is computed accurately. In order to compute K' ac- curately, it is necessary to experimentally measure e at scales where viscosity is taking energy from turbulence and transforming it from kinetic energy to heat energy. It is at such scales that most of the turbulent energy is dissipated. Viscosity becomes important at very small scales because vis- cous forces are effective only over short distances. There- fore, in order to measure e accurately, measurement must be made at very small scales, scales smaller than encountered in the inertial subrange region; that is, beyond the inertial subrange. Such difficult measurements have been made over the past few years by Grant et_ a_l (1961), Pond e_t a_l (1966) and Nasmyth (1971), as a result of care in recording, analysis and improved sensor technology. B. APPROACH Theoretically there has not been another success like Kolmgorov's inertial subrange prediction. There is no single comparable expression for the region beyond the inertial sub- range although several forms have been suggested. Because of the importance of e to other turbulence investigations, further theoretical studies at all scales of turbulence will be hampered until this small scale region is understood. This thesis, therefore, takes not a theoretical approach but an empirical approach. By using recently collected data, guideposts will be established for the theoretician by deter- mining the constants of a semi-empirical equation of the curve describing the shape of the spectrum at and beyond the iner- tial subrange region. Certain parameters will be evaluated and these evaluated parameters will form guideposts. Any theory, to be correct, should be able to reproduce the values or the empirical curve. Regression techniques will be used to determine the constants of a curve describing the shape of the spectrum of velocity fluctuations beyond the inertial sub- range region. C. KOVASZNAY'S EQUATION 1 . Theoretical Background There have been several proposals attempting to des- cribe the transfer of energy across the spectrum (Hinze, 1959, p. 190 ff ) . In terms of the three-dimensional spectrum E(k), one can write ^1^- = T(k) - 2vk2 E(k) (2) where T(k) represents, at a given wave number, the energy change due to nonlinear interactions. It is the difference between energy increase and energy decrease. When this equa- tion is averaged, phase information is lost, and as a result, there is not enough information available to solve it. The averaging, unfortunately, is absolutely necessary at some stage in the analysis. To work with the unaveraged equation means that one must know the response of the system to ini- tial and subsequent perturbations. Because of the nature of turbulent flow, this response and the perturbations are essentially unknowable. Therefore, in order to solve the 10 averaged equation, information is put in the form of physical- ly reasonable assumptions about T(k). Such proposals are usually introduced in the integral form (2) namely S(k') - / T(k) dk k' (3) The physical significance of S (k ' ) , which follows from the definition of T(k), is that it is the total flux of energy across the wave number k'. Also following from the definition of T(k) is the result T(k) = 0 (A) in the inertial subrange, because in the inertial subrange there is a local steady state. Furthermore, before the dis- sipation region is approached S(k') = e (5) the rate of turbulent energy dissipation. It is in this set- ting that we introduce the proposal of Kovasznay. 2 . Kovasznay's Proposal The essence of Kovasznay's proposal is that S(k') is determined by what goes on near k' only. In other words, he proposes that energy transfer is a local phenomenon, and is determined entirely by local parameters. From (5) and the inertial subrange law, then S(k') ~ E(k')3/2 k,5/2 = aE(k')3/2 where a is a constant. Then, from (2) (6) T(k) - tJ, S(k') k'^k _{3 KV2|E k5/2+5 E3/2k3/2} 2 9k 2 (7) 11 The negative sign is needed because k' is a lower limit; at the upper limit (k = <») , E(>) goes, to zero very strongly and so gives no contribution to TCk). In the inertial subrange so that From (7) and (9) BE QO _ Q at T(k) = 2vk E(k) 2vk2E(k)= - ia{3E1/2f k5/2+ 5E3/2 k3/2} (8) (9) (10) or simplif ied , dE dk 5 E 4 v /E/k (11) -5/3 (12) -5/3 3 k 3 a For a solution one tries E = A0k where A is a constant and 0 is an unknown function. A k is chosen since a k dependence of this nature is desired, at least in the inertial subrange. When (12) is substituted in (id || - - 5/3 A0k"8/3 + Ak"5/3 «l dk. dk Notice that 5_ E 3 k f A0k"8/3 (13) (14) relates the first terms of (13) and (11) . For the second term of (13) Ak"5/3 M. 4 V El/2 dk 3 a k f£/nr*'3* (15) 12 or r*-1/2 a* A v /A ,1/3 .. 0 d0 = --r— t fc ak ^ 3 a A Upon integrating 01/2 ■ v/A . 4/3 . . - — - k + constant 2aA Now at low wave number, from (5) so that e - aE3/2 k5/2 ... -5/3 rt.3/2 . 5/2 £ = a(Ak 0) k = aA3/2 03/2 (16) C17) (18) (19) Thus, at low wave number, 3/2 e 0' aA 3/2 or 0 1/2 _ 1/3 1/3.1/2 a A (20) which is the constant of integration occurring in (17). ,. V3 Therefore il/2 0- V_ /A 4/3 V s 2a A k .1/3 a /A (21) 3 1/4 If we introduce the Kolmogorov wave number k = (e/v ) , then 1/2 V /A t.4/3 2a A * .1/3 4/3 ax/J/I or fl _ .£.2/3 1 f 1 ,k ,.4/3,2 0 ~ (a} A [1 " TTJl (k ) ] 2a s (22) Replacing (22) in (12) eoo - (f)2/3 k"5/3 n - -* c| )4/3j2 u 2a ' s (23) 13 which is Kovasznay's formula. It is to be noted this deriva- tion leads to an equation which differs from the equation described in Hinze (1959). The appealing aspect of Kovasznay's result is that it fits observations well and is relatively easy to use. How- ever, at high wave numbers unrealistic asymptotic regions appear. Through the use of well defined spectra, the value of a will be evaluated and some precision attached to the wave number at which Kovasznay's equation is no longer valid. 3 . Application If a consistent value of a is found, then from (23) it is possible to estimate k and the shape of the spectrum beyond k . Inertial subrange measurements also allow an estimate of k from s E(k) = K' e2/3 k 5/3 (24) since K' is now reasonably well established. Various reasons exist for wanting to know the form of the velocity spectrum beyond k , but a recent one concerns laser propagation in the free atmosphere and over waves. The smallest scales of motion become very important in this application. 14 II . DESCRIPTION OF DATA A. DATA COLLECTION Six sets of data were used to obtain the spectral descrip- tions of oceanic turbulence. These data were obtained by Defense Research Establishment Pacific and analyzed by Nas- myth (1970). The data used are from his runs. Sections 1 through 5 follow closely the description given by Nasmyth (1970) in his doctoral thesis. 1 . Area These data were the results of two seagoing expedi- tions carried out in November 1967 and January-February 1969 in deep water off the west coast of Vancouver Island, British Columbia. Previously the measurements of small scale turbu- lence in the open sea by the Defense Research Establishment Pacific, Defense Research Board of Canada were made in 1962 by a submarine with hull mounted sensors. This limited the depth of measurement to about 100 meters. The sensor response permitted observations of scales down to 2 or 3 millimeters defined as "well down into the range of energy dissipation by viscous effects or up_ into the dissipation range of wave numbers." The data of 1967 and 1969 were obtained by a towed body permitting observations to depths exceeding 300 meters. The measurements are influenced by processes occurring in the upper 330 meters, of the ocean and environmental aspects of the winter months. The measurements were more or less randomly scattered over a ten thousand square kilometer area, 15 part way down the continental slope with ocean depths varying from 1000 meters hut usually over 2000 meters. The results should not be considered representative of the world oceans, but the dynamic features of the area are certainly like those in many other parts of the ocean. 2 . Sensors The sensors were mounted on the submerged body which towed from the surface ship by a servo controlled winch that compensated for ship motion to an accuracy of about 30 centi- meters in body depth in normal sea states. The winch could maintain the towed body at any desired depth down to 330 meters. In the "cycling" mode, the winch may be programmed to vary depth at a constant rate over a predetermined depth interval (typically 6-30 meters) so that at a constant towing speed the body follows a saw tooth path through the water. The velocity probe consists of an active element of evaporated platinum film on a conical glass tip. Electrical connections to the film are made through leads embedded in the glass stem. It functions like a hot wire anemometer. Since the maximum dimension of the film is approximately 0.5 millimeters, a resolution of features as small as 2 milli- meters is possible. A useful frequency response up to 1000 Hz is possible. The velocity probe is extremely sensitive to vibra- tion and extensive precautions were taken to eliminate inter- ference. Nonetheless, vibration remains the limiting factor 16 in the sensitivity of the velocity system through a midrange of the spectrum from about 1 Hz to 2Q Hz. Another major difficulty in the use of this type of probe in an oceanic environment is planktonic fouling. This is the reason all successful measurements in offshore waters have been made in 'the winter months when the plankton concen- tration is the least. When a plankton particle contacts the velocity probe, the effect is to momentarily increase the effective thickness of the boundary layer over the film. The record then shows a sharp transient indicating an apparent momentary decrease in velocity. If the plankton adhere only briefly, there is no problem, but if one is stuck, then the probe must be cleaned. This is accomplished by a system that spews a high velocity reverse flow over the tip of the probe without reducing tow- ing speed or otherwise interrupting the experiment. The velocity probe outputs are watched and the ynfouling can be accomplished in a few seconds by manual control from the ship . Under favorable conditions during the winter, a plankton particle may be contacted every few seconds, but it will only be necessary to wash the probe two or three times an hour. Therefore, it is possible to obtain records up to 30 minutes in length with only minor contamination which has negligible effect on the spectral content of the velocity signal . 17 3 . Errors The possible contamination of the velocity spectra by planktonic fouling has been discussed. The contamination of the spectra by temperature is. not so easily determined. The magnitude of possible errors was estimated by indirect methods as no direct measurements of the dynamic response of the velocity probe to fluctuations in temperature are avail- able. Without going into detail, I report that Nasmyth con- cludes that the velocity signals are at least predominantly real velocity without any major temperature component. One qualitative argument is that the power spectrum generated by the velocity signals do not exhibit any of the unique charac- teristics of the spectra of temperature fluctuations in tur- bulent flow [Grant, Hughes, Vogel, and Moilliet (1968)] and the conclusion is. that the signal must be predominantly due to velocity fluctuations. 4 . Spectra The power spectra have been computed from a number of samples of velocity fluctuations recorded during the November 1967 and January-February 1969 cruises. For satisfactory spectral analysis, it is necessary to work with reasonably uniform samples of as long a duration as possible. Besides the problems already discussed, there are other interferences that are frequently encountered in the data collection. Tur- bulence occurs in patches within layers. Since there is no way of predicting in advance what the signal levels in the next patch will be, many good samples are lost because of 18 inappropriate gain settings. When probe washes are necessary it may ruin an otherwise useful section of record. Any ap- preciable change in ship's speed during recording will dis- tort the resulting spectrum. The point is that if one is able to get suitable samples of any length, the sample is valuable . During November 1967, 32 hours of recording during two weeks at sea were accomplished. During January-February 1969, 35 hours of recording during three weeks at sea were made. From these raw data samples, there are 20 or 25 samples varying in length from 10 seconds to two minutes from which meaningful spectra were derived. In the cycling mode sample, times are short because of changing turbulence layers. In the constant depth mode sample, lengths are longer and several spectra of 52 seconds duration or longer were obtained Wherever possible, noise is subtracted from the original spectrum resulting in a noise-free spectrum. 5 . Recording Components of circuitry which must be located close to the sensors are enclosed in pressure cases in the towed body. From there, information from all sensors is fed up the armored multi-conductor towing cable for further processing and recording on the ship. The lower 100 meters of cable are faired to reduced vibration. On board ship the signal from each thermistor is divided into two channels. One is amplified and recorded directly and the other is electronically differentiated with 19 respect to time before recording. This method gives an ac- ceptable signal-noise ratio at the higher frequencies other- wise unobtainable on a single channel. The signal from the velocity probe is divided into three channels for similar reasons. The first channel or "A" channel is recorded broad- band with no filtering. "B" channel is effectively differ- entiated up to about 10 Hz and cuts off at about 20 Hz. "C" channel differentiates up to 1000 Hz and then cuts off. The signal from the temperature probe is treated similarly except that there is a low-frequency cutoff on "A" at about 0.01 Hz and "B" extends up to about 100 Hz. Signal channels from all sensors together with coded timing signals and a voice commentary are recorded on two seven channel magnetic tapes with multiplexing as required. At the same time., selected channels are monitored on oscillo- scopes as an aid in selecting appropriate gain settings and 12 channels are recorded on two paper charts giving a perman- ent visual record. This served as a planning aid during the experiment and for later use during analysis. B. DATA USED Data sets 4, 5, and 6 consisted of 45 to 47 data points; whereas, data sets. 7, 8 and 9 consisted of 174 to 176 data points. These six sets of data were plotted on log-log graph 0 (k) k paper in normalized form as log — ^-~ — —jj- vs log (r- ) , (Figures (ev ) s 1-6). The data are presented in this form to make comparison of spectra easier. Normalized plots should lie on top of each 20 other. The plots covered 8 to 11 decades along the 0(k), or energy axis and 4 to 6 decades along the k, or wave number axis. This remarkably large range of values makes them ideally suited for examining details of the high wave number region of the turbulent energy spectrum and testing Kovasznay's equation . The data points clearly indicated the expected -5/3 slope inertial subrange region and the "tailing off" in the low energy, high wave number region beyond the inertial subrange. Several points were discarded by this author because of ob- vious inconsistencies. Only runs 4, 5, and 6 had no points discarded. Run 7 had eleven points discarded; 8 had ten points discarded and run 9 had nineteen points. Because of the similarity of these data sets, it was considered necessary only to use data set 4 for analysis. 21 III. ANALYSIS AND APPROACHES A. APPROACHES 1 . Linear Regression Since a direct substitution of a for (23) is diffi- cult, the equation was expressed in a form which was amenable to regression techniques. Despite several attempts, consis- tent results could not be obtained and this method of approach to the problem was discarded. 2 . Inertial Subrange An alternate approach was to consider only the ISR. In the inertial subrange the one-dimensional energy spectrum and the three-dimensional energy spectrum have the same power law behavior. The Kovasznay equation then may be reduced to 000 * a"2/3 e2/2 k"5/3 (25) [where 0(k) is the one-dimensional energy equation] since clearly these terms will dominate. This reduced form allows observed 0 (k) and k to be used in this region to evaluate a. Since 0(k) = K' £2/3 k"5/3 (26) -2/3 then it is natural to associate a with K'. Since K' - 0.5, then an a value of 1.59 can be expected. 3 . Exponential Curve Beyond the inertial subrange, 0 (k) falls off expon- entially with k. A desirable equation would be one that collapsed to Equation (26) in the inertial subrange but 22 generated an exponential decay at high wave numbers. Further, such an equation should be expressed in terms of meaningful physical quantities. This is the goal of the theoretician if not this thesis. The object here is simply to provide empirical clues. Consequently, an exponential regression analysis was attempted. B. ANALYSIS 1 . Linear Regression Equation (23) was taken and rewritten in the form r v ' s s (27) Simplified into an equation suitable for regression techniques this becomes Y = A.^""5/3 + A2k~1/3 + A3k (28) where A.., A„ and A„ are given below. The regression analysis program appeared to be very sensitive to bad data. According to the Kovasznay equation, A, and A should be > 0 and A_ < 0. Furthermore, | A.. | > | A„ | > | A_ | . No such consistency was found for the six sets of data when using the Regre Program. Of the six sets, there were in fact three pairs. Each pair corresponded to spectra from a single time series of turbulence data. The difference is that one set of the pair contained four times as many points as the other. Even in the paired data sets, results proved inconsistent. Further, the poor regeneration of the 23 curve using the A,, A„ , and A., computed by Regre caused this method to be discarded. 2 . Subs titut ion Equation C28) was used where 2/3 Aj = (e/a) 2/3 A2 - -e Caks) 4/3 2/3 .25e ' A3 2. 8/3 a k A direct substitution of a = 1.6 was used. The values of A,, A„, and A„ were computed. The values are presented in Table I. These values give a regeneration of the curve and is an alternative but simpler way of expressing Kovasz- nay's equation. A comparison of the curve generated with these values of A,, A„ , and A„ and actual data is shown in Figure 7. The agreement is good in the inertial subrange. 3 . Gompertz Curve The Gompertz curve y = e was used because it was readily available. It is unsatisfactory, however, be- cause even if solved and a good curve fit obtained, the quan- tities a, 3, P, and x do not lend themselves to physical interpretation. There are no dimensions on the right hand side. Another curve should be used but the form is not imme- diately obvious and is perhaps worthy of a separate research inves tigation . 24 IV. CONCLUSIONS Since approaches III-A-1 and III-A-3 were unsuccessful, only the result of Approach III-A-2 will be discussed. As mentioned in III-A-2, an a value of 1.59 was expected. A spread of a from 1.8 to more than 6.0 was obtained from the real data in the manner described below. A plot was made of a versus wave number (Figure 8) . A minimum occurs in the vicinity of k " 0.5. Beyond this wave number a, values continue to increase with increasing wave number . Various values of a obtained using the second method above were substituted in the full Kovasznay equation and the resulting curves compared with actual data (Figure 9). Of the several curves generated, low values of a generated better fits, the best fit occurring for the lowest a, namely a = 1.8. This is also closest to the expected value of a - 1.6. The fit of curves with a = 1.6 and a = 1.8 were very similar although the lower a value curve dropped off more steeply. Another feature of interest is that in the inertial subrange, 0(k) values are lower for higher a (since a is raised to a negative power) but beyond the inertial subrange (where other terms dominate) the opposite is true. The crossover point occurs around k - 20. The significance of this, If any , is not clear . 25 The limit of the Koyasznay result appears to be at a wave number k - 50. Beyond this wave number, the values of energy increase with increasing wave number, a phenomenon that was expected. This phenomenon obviously is inconsistent with ob- served results and indeed would be catastrophic theoretically. However, practically it is not important since present tech- nology does not allow measurements to be made at such high wave numbers . In summary, as a increases, the fitted curve approximates the actual curve with decreasing accuracy. The best a occurs at the minimum in the a versus k (Figure 8) and is approxi- mately where k < 0.5. The reconstructed curve agrees well up to k - 2.0. For k > 2.0, 0 (k) in actual data falls off more rapidly than 0(k) computed. Good agreement in the ISR is expected and observed. The dissipation range was not expected to agree. Figure 9 displays the quantitative results showing existing agreement. 26 TABLE I A p 0.3753922 A2 ^-0.0056287 A3 = 0.0000211 27 APPENDIX A Velocity Spectrum Data Set 4 .2266+01 .4533+01 .6800+01 .9066+01 .1133+02 .1360+02 .1586+02 .2040+02 .2493+02 .2946+02 .9070+00 .1813+01 .2720+01 .3626+01 .4533+01 .5440+01 .6346+01 .8160+01 .9973+01 .1178+02 .3630+00 .7250+00 .1088+01 .1450+01 0 00 .4364-01 .8166-02 .2103-02 .5977-03 .2055-03 .7826-04 .3360-04 .7937-05 .2734-05 .1177-05 .2542+00 .8550-01 .3687-01 .1682-01 .8331-02 .4598-02 .2607-02 .8691-03 .3628-03 .1596-03 .1615+01 .3830+00 .1911+00 .1229+00 .1813+01 .2538+01 .3264+01 .3989+01 .1360+00 .2040+00 .2720+00 .3400+00 .4760+00 .6120+00 .7480+00 .8840+00 .1020+00 .0544+00 .0680+00 .0816+00 .0952+00 .1224+00 .1496+00 .1768+00 .2040+00 .2312+00 .3268+00 0 00 8377-01 4508-01 2399-01 1258-01 7338+01 3987+01 2511+01 1899+01 1163+01 6428+00 4442+00 3645+00 2606+00 3321+02 2045+02 1726+02 1360+02 9453+01 6308+01 6341+01 4608+01 3297+01 1945+01 28 APPENDIX B Velocity Spectrum Data Set 5 0 00 0 00 .4327+01 .6490+01 .8658+01 .1081+02 .1298+02 .1514+02 .1730+02 .1947+02 .2380+02 .2812+02 .1730+01 .2596+01 .3461+01 .4327+01 .5192+01 .6058+01 .7789+01 .9520+01 .1125+02 .6923+00 .1038+01 .1384+01 .1730+01 .1264-01 .3475-02 .1007-02 .3460-02 .1313-03 .5580-04 .2539-04 .1225-04 .3536-05 .1302-05 .1174+00 .5373-01 .2606-01 .1346-01 .7589-02 .4347-02 .1497-02 .6147-03 .2664-03 .4671+00 .2467+00 .1682+00 .1168+00 2423+01 3115+01 3808+01 1298+00 1947+00 2596+00 3245+00 4543+00 5841+00 7140+00 8438+00 9736+00 5192-01 6490-01 7789-01 9087-01 1168+00 1428+00 1687+00 1947+00 2206+00 3115+00 6419-01 3671-01 2015-01 9699+01 5546+01 3611+01 2408+01 1268+01 7726+00 5618+00 4313+00 3431+00 4939+02 3352+02 2398+02 1841+02 1176+02 8556+01 7468+01 5946+01 5076+01 2790+01 29 APPENDIX C Velocity Spectrum Data Set 6 0GO 0 00 .4501+01 .6076+01 .8102+01 .1012+02 .1215+02 .1417+02 .1620+02 .1822+02 .2228+02 .2633+02 .1620+01 .2430+01 .3240+01 .4051+01 .4861+01 .5671+01 .7291+01 .8912+01 .1053+02 .6481+00 .9722+00 .1296+01 .1696+01 .1518-01 .4249-02 .1259-02 .4279-03 .1634-03 .7195-04 .3287-04 .1597-04 .4921-05 .2059-05 .1547+00 .6958-01 .3245-01 .1676-01 .9774-02 .5748-02 .2027-02 .8019-03 .3577-03 .5820+00 .3049+00 .2094+00 .1449+00 .2268+01 .2916+01 .3564+01 .1215+00 .1822+00 .2430+00 .3038+00 .4253+00 .5468+00 .6684+00 .7899+00 .9104+00 .4861-01 .6076-01 .7291-01 .8507-01 .1093+00 .1336+00 .1579+00 .1882+00 .2066+00 .2916+00 .8186-01 .4336-01 .2392-01 .1328+02 .7371+01 .4531+01 .3101+01 .1678+01 .1075+01 .6782+01 .5269+01 .4223+01 .4705+02 .3137+02 .2536+02 .2171+02 .1421+02 .1054+02 .6388+01 .6400+01 .4356+01 .2565+01 30 APPENDIX D Velocity Spectrum Data Set 7 000 .2163+01 .4327+01 .6490+01 .8654+01 .1081+02 .1298+02 .1514+02 .1730+02 .1947+02 .2163+02 .2380+02 .2596+02 .2812+02 .3029+02 .3245+02 .3461+02 .3678+02 .3894+02 .8554+00 .1730+01 .2596+01 .3461+01 .4327+01 .5192+01 .6058+01 .6923+01 .7789+01 .8654+01 .9520+01 .1038+02 .1125+02 .1211+02 .1298+02 .1384+02 .3461+00 .6923+00 .1038+01 .1384+01 .1730+01 .2077+01 .2423+01 .2769+01 .3115+01 .3461+01 6141-01 1264-01 3475-02 1007-02 3460-03 1313-03 5580-04 2539-04 1225-04 6396-05 3536-05 2066-05 1302-05 8278-06 5645-06 4053-06 3089-06 2456-06 3147+00 7174+00 5373-01 2606-01 1346-01 7589-02 4347-02 2526-02 1497-02 9408-03 6147-03 4032-03 2664-03 1798-03 1237-03 7531-04 1936+01 4671+00 2467+00 1682+00 1168+00 8659-01 6419-01 4715-01 3671-01 2789-01 .3808+01 .4154+01 .4500+01 .4846+01 .5192+01 .3461+00 .6923+00 .1038+01 .1384+01 .1730+01 .2077+01 .2423+01 .2769+01 .3115+01 .3461+01 .3808+01 .4154+01 .6490-01 .1298+00 .1947+00 .2596+00 .3245+00 .3894+00 .4543+00 .5192+00 .5841+00 .6490+00 .7140+00 .7789+00 .8438+00 .9087+00 .9736+00 .1038+01 .1103+01 .1163+01 .1298+01 .1428+01 .1557+01 .1687+01 .1817+01 .1947+01 .2077+01 .1298-01 .2596-01 0(10 2015-01 1511-01 1183-01 9733-02 7448-02 2533+01 5638+00 2937+00 1935+00 1317+00 9446-01 6852-01 4991-01 3887-01 2913-01 2096-01 1532-01 1722+02 9699+01 5546+01 3611+01 2404+01 1659+01 1268+01 1015+01 7726+00 6376+00 5618+00 4778+00 4313+00 4004+00 3431+00 2916+00 2697+00 2479+00 2317+00 1788+00 1693+00 1395+00 1176+00 1150+00 4810-01 1287+04 2217+03 31 .3894-01 .5192-01 .6490-01 .7789-01 .9087-01 .1038+00 .1168+00 .1298+00 .1428+00 .1557+00 .1687+00 .1817+00 .1947+00 .2077+00 .2206+00 .2336+00 .2596+00 .2856+00 .3115+00 .3375+00 .3634+00 .3894+00 .4154+00 .1298-01 .2596-01 .3894-01 .5192-01 .6490-01 .7789-01 .9087-01 .1038+00 .1168+00 .1298+00 .1428+00 .1557+00 .1687+00 .1817+00 .1947+00 .2077+00 .2206+00 .2336+00 .2596+00 .2856+00 .3115+00 .3375+00 .3634+00 .3894+00 .4154+00 -.1886+01 -.1585+01 -.1409+01 -.1284+01 0(k) .7043+02 .5097+02 .3334+02 .2404+02 .1796+02 .1425+02 .1067+02 .8304+01 .7412+01 .7218+01 .6513+01 .5709+01 .5139+01 .4900+01 .4386+01 .3600+01 .3771+01 .3081+01 .2636+01 .2099+01 .1674+01 .1615+01 .1615+01 .5325+03 .1245+03 .6853+02 .4939+02 .3352+02 .2398+02 .1841+02 .1550+02 .1176+02 .9237+01 .8556+01 .8029+01 .7468+01 .6568+01 .5946+01 .5624+01 .5076+01 .4014+01 .4225+01 .3275+01 .2790+01 .2229+01 .1762+01 .1642+01 .6738+00 .2726+01 .2095+01 .1835+01 .1693+01 1187+01 1108+01 1041+01 9835+00 9324+00 8866+00 8452+00 8074+00 7727+00 7405+00 7105+00 6825+00 6562+00 6313+00 5856+00 5442+00 5064+00 4716+00 4395+00 4095+00 3815+00 1886+01 1585+01 1409+01 1284+01 1187+01 1108+01 1041+01 9835+00 9324+00 8866+00 8452+00 8074+00 7727+00 7405+00 7105+00 6825+00 6562+00 6313+00 5856+00 5442+00 5064+00 4716+00 4395+00 4095+00 3815+00 1081-02 2163-02 3245-02 ,4327-02 5409-02 ,6490-02 0(k) .1525+01 .1379+01 .1265+01 .1190+01 .1070+01 .9655+00 .9322+00 .9046+00 . 8732+00 .8174+00 .7742+00 .7500+00 .7055+00 .6036+00 .6258+00 .5153+00 .4456+00 .3482+00 .2460+00 .2155+00 .1714+00 .3109+01 .2345+01 .1847+01 .1707+01 .1523+01 .1381+01 .1254+01 .1153+01 .1028+01 .9193+00 .8699+00 .8584+00 .8137+00 . 7566+00 .7108+00 .6902+00 .6421+00 .5563+00 .5764+00 .4887+00 .4209+00 .3220+00 .2237+00 .2082+00 .1000+02 .9 721+04 .6339+04 .4608+04 .3343+04 .2598+04 .1879+04 32 7572-02 8654-02 9736-02 1081-01 1190-01 1298-01 1406-01 1514-01 1622-01 1730-01 1839-01 1947-01 2163-01 2380-01 2596-01 2812-01 3029-01 3245-01 3461-01 3678-01 3894-01 4110-01 4327-01 4760-01 5192-01 5625-01 6058-01 6490-01 6923-01 7356-01 7789-01 8221-01 8654-01 .1271+04 .1207+04 .7168+03 .3742+03 .5903+03 .5717+03 .3760+03 .3715+03 .3424+03 .3385+03 .6875+03 .3491+03 .1060+03 .9513+02 .9 774+02 .1222+03 .6649+02 .5879+02 .5220+02 .5370+03 .8703+02 .3341+03 .4864+02 .2585+02 .4629+02 .5433+02 .4407+02 .3166+02 .3758+02 .2539+02 .2104+02 .2558+02 .2025+02 33 APPENDIX E Velpcity Spectrum Data Set 0 00 0 00 2025+01 4051+01 6076+01 8102+01 1012+02 1215+02 1A17+02 1620+02 1822+02 2025+02 2228+02 2430+02 2636+02 2635+02 3038+02 3240+02 3443+02 3645+02 ,8102+00 ,1620+01 ,2430+01 ,3240+01 ,4051+01 ,4861+01 ,5671+01 ,6481+01 ,7291+01 .8102+01 ,8912+01 ,9722+01 .1053+02 .1134+02 .1215+02 .1296+02 .3240+00 .6481+00 .9722+00 .1296+01 .1620+01 .1944+01 .2268+01 .2592+01 .2916+01 .3240+01 .3504+01 .7442-01 .1518-01 .4249-02 .1259-02 .4279-03 .1634-03 .7145-04 .3287-04 .1597-04 .8187-05 .4921-05 .2787-05 .2059-05 .1159-05 .9579-06 .5871-06 .5260-06 .3692-06 .4143+00 .1547+00 .6958-01 .3245-01 .1676-01 .9774-02 .5748-02 .3409-02 . 2027-02 .1251-02 .8019-03 .5324-03 .3577-03 .2383-03 .1624-03 .9926-04 .2457+01 .5820+00 .3049+00 .2094+00 .1449+00 .1076+00 .8186-01 .6005-01 .4336-01 .3140-01 .2392-01 3889+01 4213+01 4537+01 4861+01 6076-01 1215+00 1822+00 2430+00 3038+00 3645+00 4253+00 4861+00 5468+00 6076+00 6684+00 7291+00 7899+00 ,8507+00 ,9114+00 ,9 722+0 0 ,1033+01 ,1093+01 ,1215+01 ,1336+01 ,1458+01 ,1579+01 .1701+01 ,1822+01 ,1944+01 .3240+00 .6481+00 .9722+00 .1296+01 .1620+01 .1944+01 .2268+01 .2592+01 .2916+01 .3240+01 .3634+01 .3889+01 .1215-01 .2430-01 .3645-01 .4861-01 .1863-01 .1431-01 .1190-01 .9555-02 .2395+02 .1328+02 .7371+01 .4531+01 .3101+01 .2290+01 .1678+01 .1334+01 .1075+01 .8430+00 .6782+00 .5776+00 .5269+00 .4803+00 .4223+00 .3648+00 .3426+00 .3213+00 .2948+00 .2458+00 .212 7+00 .1674+00 .1740+00 .1280+00 .1301+00 .3383+01 .7058+00 .3638+00 .2421+00 .1847+00 .1180+00 .8795-01 .6393-01 .4561-01 .3265-01 .2467-01 .1870-01 .6753+03 .1545+03 .7864+02 .4705+02 34 ,6076-01 .7291-01 .8507-01 .9722-01 .1093+00 .1215+00 .1336+00 .1458+00 .1579+00 .1701+00 .1822+00 .1944+00 .2086+00 .2187+00 .2430+00 .2916+00 .3159+00 .3402+00 .3645+00 .3889+00 .1215-01 .2430-01 .3645-01 .4861-01 .6076-01 .7291-01 .8507-01 .9722-01 .1093+00 .1215+00 .1336+00 .1458+00 .1579+00 .1701+00 .1822+00 .1944+00 .2066+00 .2187+00 .2430+00 .2673+00 .2916+00 .3159+00 0 00 .3137+02 .2536+02 .2171+02 .1786+02 .1421+02 .1235+02 .1054+02 .8478+01 .7336+01 .7237+01 .7342+01 .5975+01 .4964+01 .4904+01 .4309+01 .2570+01 .2240+01 .2224+01 .1627+01 .1782+01 .2404+04 .2778+03 .7597+02 .4960+02 .2984+02 .2551+02 .2115+02 .1725+02 .1304+02 .1155+02 .9539+01 .7767+01 .6388+01 .6289+01 .6400+01 .5371+01 .4356+01 .4443+01 .3855+01 .2962+01 .2565+01 .2171+01 .3402+00 .3645+00 .3889+00 .1012-02 .2025-02 .3038-02 .4051-02 .5063-02 .6076-02 .7089-02 .8102-02 .9114-02 .1012-01 .1114-01 .1215-01 .1316-01 .1417-01 .1519-01 .1620-01 .1721-01 .1822-01 .2025-01 .2228-01 .2430-01 .2633-01 .2835-01 .3038-01 .3240-01 .3443-01 .3645-01 .3848-01 .4051-01 .4456-01 .4861-01 .5266-01 .5671-01 .6076-01 .6481-01 .6886-01 .7291-01 .7697-01 .8102-01 0 00 .2195+01 .1592+01 .1764+01 .2081+05 .1359+05 ,8952+04 .5749+04 .5161+04 .4043+04 .2057+04 .1260+04 .8931+03 .5926+03 .4505+03 .4141+03 .3837+03 .2777+03 .2459+03 .2012+03 .1791+03 .2685+03 .1747+03 .1572+03 .1538+03 .1735+03 .9218+02 .1327+03 .7887+02 .1136+03 .83 7 4+0 2 .7422+02 .5865+02 .2869+02 .3091+02 .2391+0 2 .4713+02 .3175+02 .1761+02 .3178+02 .1023+02 .2331+02 .2476+02 35 APPENDIX F Velocity Spectrum Data Set 9 0 00 0 00 .2266+01 .4533+01 .6800+01 .9066+01 .1133+02 .1360+02 .1586+02 .1813+02 .2040+02 .2266+02 .2493+02 .2720+02 .2946+02 .3173+02 .3400+02 .3626+02 .3853+02 .4080+02 .9066+00 .1813+01 .2720+01 .3626+01 .4533+01 .5440+01 .6346+01 . 7253+01 .8160+01 .9066+01 .9975+01 .1089+02 .1178+02 ,1269+02 .1360+02 .1450+02 .3626+00 .7253+00 .1038+01 .1450+01 .1813+01 .2176+01 .2538+01 .2901+01 .3264+01 .3826+01 .3989+01 .4364-01 .8166-02 .2103-02 .5977-03 .2055-03 .7826-04 .3360-04 .1567-04 . 7937-05 .4434-05 .2734-05 .1797-05 .1177-05 .8137-06 .5987-06 .4416-06 .3509-06 .2922-06 .2452+00 . 8550-01 .3687-01 .1682-01 .8331-02 .4598-02 .2607-02 .1492-02 . 8691-03 .5539-03 .3528-03 .2381-03 .1596-03 .1055-03 .7012-04 .4286-04 .1615+01 .3830+00 .1911+00 .1229+00 .8377-01 .6140-01 .4508-01 .3290-01 .2399-01 .1692-01 .1258-01 .4352+01 .4714+01 .5077+01 .5440+01 .3626+00 .7253+00 .1088+01 .1450+01 .1813+01 .2176+01 .2538+01 .2901+01 .3264+01 .3626+01 .3989+01 .4352+01 .2833-01 .5666-01 .6800-01 .1360+00 .2040+00 .2720+00 .3400+00 .4080+00 .4760+00 .5440+00 .6120+00 .6800+00 .7480+00 .8160+00 .8840+00 .9520+00 .1020+01 .1088+01 .1156+01 .1224+01 .1360+01 .1496+01 .1632+01 .1768+01 .1904+01 .2040+01 .2176+01 .1360-01 .2720-01 .9581-02 .7259-02 .5953-02 .4404-02 .2190+01 .4729+00 .2351+00 .1502+00 .1030+00 .7319-01 .5268-01 .3860-01 .2789-01 .1921-01 .1437-01 .1086-01 .1086-01 .1086-01 .1357+02 .7338+01 .3987+01 .2511+01 .1899+01 .1524+01 .1163+01 .8491+00 .6428+00 .5384+00 .4442+00 .3873+00 .3645+00 .3071+00 .2606+00 .2463+00 .2169+00 .1714+00 .1668+00 .1471+00 .1150+00 .1036+00 .9932-01 .9028-01 .6476-01 .5393+03 .1416+03 36 0 00 0Ck) .4080-01 .5440-01 .6800-01 .8160-01 .9520-01 .1088+00 .1224+00 .1360+00 .1496+00 .1832+00 .1763+00 .1904+00 .2040+00 .2176+00 .2312+00 .2448+00 .2720+00 .2992+00 .3264+00 .3536+00 .3808+00 .4080+00 .4352+00 .1360-01 .2720-01 .4080-01 .5440-01 .6800-01 .8160-01 .9520-01 .1088+00 .1224+00 .1360+00 .1496+00 .1632+00 .1768+00 .1904+00 .2040+00 .2176+00 .2312+00 .2448+00 .2720+00 .2992+00 .3264+00 .6473+02 .3321+02 .2045+02 .1726+02 .1360+02 .1119+02 .9453+01 .7766+01 .6308+01 .5954+01 .6341+01 .5562+01 .4608+01 .3920+01 .3297+01 .2555+01 .2475+01 .2507+01 .1945+01 .1544+01 .2011+01 .1421+01 .1376+01 .9109+03 .2283+03 .7399+02 .3535+02 .2012+02 .1635+02 .1399+02 .1113+02 .9165+01 .7516+01 .6018+01 .5731+01 .6061+01 .5376+01 .4234+01 .3527+01 .2933+01 .2334+01 .2147+01 .2400+01 .1889+01 3536+00 3808+00 4Q80+00 4352+00 1133-02 2266-02 3400-02 4533-02 5666-02 6800-02 7933-02 9066-02 1020-01 1133-01 1246-01 1360-01 1473-01 1586-01 1700-01 1813-01 1926-01 2040-01 2266-01 2493-01 2720-01 2946-01 3173-01 3400-01 3626-01 3853-01 4080-01 4306-01 4533-01 4986-01 5440-01 5893-01 6346-01 6800-01 7253-01 7703-01 8160-01 8613-01 9066-01 .1470+01 .2018+01 .1462+01 .1413+01 .5631+04 .3842+04 .2843+04 .2069+04 .1862+04 .1248+04 .7569+03 .7216+03 .5015+03 .2628+03 .1317+03 .1596+03 .4404+03 .6856+03 .7990+03 .6287+03 .2860+03 .1644+03 .1902+03 .2100+03 .1512+03 .1310+03 .1027+03 .7257+02 .5452+02 .5672+02 .6259+02 .6576+02 .5946+02 .6716+02 .1814+02 .2297+02 .1206+02 .1747+02 .2328+02 .2080+02 .1705+02 .1192+02 .2965+02 37 in 60 o 4- 3- 2- 1- Figure 1 Velocity Spectrum For Data Set 4 o o 1-1 -2- -3- -3 LOG J (cgs) 38 to 60 O o o 4- 3- 2-1 0- -1- -2- ■3- Figure 2 Velocity Spectrum For Data Set 5 "=T ! -1 LOG £ (cgs) 39 60 O IS o V Figure 3 Velocity Spectrum For Data Set 6 2- r. 1- -2- -3- -4- LOG r -1 (cgs) 40 . v :f 3- s Figure A Velocity Spectrum For Data Set 7 2- 1- w 00 o 0- e> o -2- -3-! -4- 2 k LOG ^ (cga) 41 3- 2-i 1-i • t Figure 5 Velocity Spectrum For Data Set 8 •V M H o 0- -2- -3- -4- -t LOG | (cgs) 42 3- Figure 6 Velocity Spectrum For Data Set 9 2^ in M o 0- o o -2- -3- -1 LOG | (cgs) s 43 en 60 o 1- 0- -1- -2~ *.A • • .A Figure 7 • Actual Non-normalized data A Curve Regeneration using A , A , A values A :.a .A •A ** .A A • A o o ►J -3- . A -4- -5- LOG k (cgs) 44 M co >. 0) CO CO * o> Ci 3 e u N V) 3 3 CO H C 00 (fl 0) •H P> > 01 Ph o > ^ 91 OG y o -r o T o io XeuzsBAO)] 45 A B Figure 9 w u O O -1-1 -2-1 -3- Curve Regeneration Using Actual a Values <]•□ A a = 1.59 • a = 1.8 □4 □ a = 4 .0 □A. Qa Q '4 □ 6 '4 4 & ■4- -5- -6- -i LOG k (cgs) A °a Q 46 BIBLIOGRAPHY 1. Batchalor, G. K., The Theory of Homogeneous Turbulence, Cambridge University Press, 1953. 2. Grant, H. L . , and Molliet, A., "The Spectrum of a Cross Stream Component of Turbulence in a Tidal Stream," J_. Fluid Mech. , 13, 237, 1962. 3. Grant, H. L., Hughes, B. A., Vogel, W. M., and Molliet, A., "The Spectrum of Temperature Fluctuations in Turbu- lent Flow," J. Fluid Mech. , 34 , 423, 1968. 4. Hinze, J. 0., Turbulence , McGraw-Hill Book Company, 1959. 5. Kolmogorov, A. N., "The Local Structure of Turbulence in Incompressible Viscous Fluid for very Large Reynolds Numbers," C. R. Acad. Sci. U.R.S .S . , 30, 301, 1941. 6. Kolmogorov, A. N., "A Refinement of Previous Hypothesis Concerning the Local Structure of Turbulence in a Viscous Incompressible Fluid at High Reynolds Number," J_. Fluid Mech. , 13, 82, 1962 .' 7. Nasmyth, P. W., Oceanic Turbulence, Ph.D. thesis, University of British Columbia, Vancouver, 1970. 8. Oboukhov, A. M., "Some Specific Features of Atmosphere Turbulence," J. Fluid Mech. , 13, 77, 1962. 9. Stewart, R. W., and Grant, H. L., "Determination of the Rate of Dissipation of Turbulent Energy Near the Sea Surface in the Presence of Waves," J. Geophy s . Res. , 6 7 , 3177, 1962. 47 INITIAL DISTRIBUTION LIST 1. 2. 3. 4. 5. 6. 7. 8. Defense Documentation Center Cameron Station Alexandria, Virginia 22314 Library, Code 0212 Naval Postgraduate School Monterey, California 93940 Professor N. E. Boston, Code 58Bb Department of Oceanography Naval Postgraduate School Monterey, California 93940 Professor Kenneth L. Davidson, Code 51Ds Department of Meteorology Naval Postgraduate School Monterey, California 93940 No . Copies 2 Lieutenant Commander Richard B. 5001 Rebel Trail N.W. Atlanta, Georgia 30327 Belser III Department of Oceanography, Code 5 Naval Postgraduate School Monterey, California 93940 Oceanographer of the Navy The Madison Building 732 N. Washington Street Alexandria, Virginia 22314 Dr. Ned A. Ostenso Code 480D Office of Naval Research Arlington, Virginia 22217 48 Security Classification DOCUMENT CONTROL DATA -R&D [Security c las si lication ol title, body ol abstract and indexing annotation must be entered when the overall report ts classified) I ORIGINATING ACTIVITY (Corporate author) Naval Postgraduate School Monterey, California 9394Q 2«. REPORT SECURITY CLASSIFICATION Unclassified 2b. GROUP 3 REPORT TITLE Empirical Description of High. Wave Number Turbulence Spectra 4. DESCRIPTIVE NOTES (Type ol report and,inclusive dates) Master's Thesis; September 1972 5. au THOR1SI (First name, middle initial, last name) Richard B. Belser III 6- REPOR T D A TE September 1972 7a. TOTAL NO. OF PAGES 50 7b. NO. OF REFS Sa. CONTRACT OR GRANT NO. b. PROJEC T NO. 9a. ORIGINATOR'S REPORT NUMBER(S) 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) 10. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Postgraduate School Monterey, California 93940 13. ABSTR AC T The shapes of spectra obtained from measurements of ocean turbulence are examined. Kovasznay's equation for high wave number turbulence is used to describe the inertial through dissipation regions. Power spectra of velocity fluctuation recorded in the open sea are used to evaluate constants occur- ring in the Kovasznay relation. Semi-empirical forms of high wave number energy spectra are then displayed. These computed spectra should define a universal form to which mathematical models should adhere. DD 1 NOV 68 • *V I *J S/N 0101 -807-681 1 (PAGE 1 ) 49 Security Classification i-SUOJ Security Classification key wo ROS Dissipation region Inertial subrange Isotropic turbulence Kovasznay equation Linear regression techniques Power spectra Wave number DD ,F°r„1473 S/N 0101-807-682 1 BACK) 50 Security Classification A- 3 1 I ftp ,' ~ • Of L . r'Cv» I sPec t/v Jer on tor. ^ Thesis .135466 B3653 Belser c." Empirical description of high wave number tur- bulence spectra. thesB3653 EZK descr|Ptlon of high wave numbe 3 2768 002 13006 4 DUDLEY KNOX LIBRARY